The Nth-order bright and dark solitons for the higher-order nonlinear Schrödinger equation in an optical fiber

Under investigation in this paper is a sextic nonlinear Schrödinger equation, which can describe the attosecond pulses in an optical fiber. The Nth-order bright and dark soliton solutions are derived via the Hirota method. We demonstrate that the interaction between the two solitons is elastic unless the solitons propagate with the same velocities. Meanwhile, mathematical expressions of the amplitude, velocity and phase shift of each soliton are given under five cases. We reveal that two interactive bright solitons can evolve to the parallel-state solitons or bound solitons when the two solitons have the same velocities while the interaction of the dark solitons can not generate the bound solitons. Furthermore, if two branches of the second-order solitons possess the identical wave numbers and angular frequencies, the second-order solitons will degenerate into the first-order soliton.